Problem 3 - Applied Calculus

Question

Integration by parts often involves finding integrals like the following when integrating \(d v\) to find \(v\). Find the following integrals without using integration by parts (using formulas 1 through 7 on the inside back cover). Be ready to find similar integrals during the integration by parts procedure. \(\int(x+2) d x\)

Step-by-Step Solution

Answer

\( \int (x+2) \ dx = \frac{x^2}{2} + 2x + C \).

Step 1: Identify the Integral Form

The given integral is \( \int (x+2) \ dx \). This can be split into two separate integrals using the linearity of integration: \( \int x \ dx + \int 2 \ dx \).

Step 2: Integrate Each Term Separately

Using basic integration rules, integrate each term separately. For \( \int x \ dx \), use the power rule: \( \int x^n \ dx = \frac{x^{n+1}}{n+1} + C \) for \( n = 1 \). This gives \( \frac{x^2}{2} + C \). For \( \int 2 \ dx \), use the constant rule: \( \int a \ dx = ax + C \), which yields \( 2x + C \).

Step 3: Combine the Results

Add the results from the two integrals: \( \frac{x^2}{2} + 2x + C \). This is the result of the given integral \( \int (x+2) \ dx \).

Key Concepts

Integration by Parts

Integration by parts is a powerful method used to tackle integrals that are difficult to solve directly. It is especially helpful when dealing with products of functions. The principle behind integration by parts is based on the product rule for differentiation.
The formula is given by:

• \( \int u \, dv = uv - \int v \, du \)

where \( u \) and \( dv \) are parts of the original integrand.
Choosing \( u \) and \( dv \) properly is crucial to simplifying the integral. Often, selecting \( u \) as a function that simplifies upon differentiation, while \( dv \) is easily integrable, works best.
In the problem you tackled, although you could potentially use integration by parts, there were simpler ways due to the nature of the integrand, \( x + 2 \). This highlights an important skill in calculus: identifying when a more straightforward rule can be more efficient.

Linearity of Integration

Linearity of integration is a fundamental concept that simplifies the integration process. It refers to the ability to split an integral into a sum or difference of separate integrals.
The principle can be stated as follows:

• \( \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \)
• \( \int (f(x) - g(x)) \, dx = \int f(x) \, dx - \int g(x) \, dx \)

This property makes it easier to manage complex expressions by breaking them into simpler parts that are often more straightforward to evaluate.
In your exercise, you applied this rule to separate \( \int (x+2) \, dx \) into two integrals: \( \int x \, dx + \int 2 \, dx \), allowing them to be handled independently.

Power Rule in Integration

The power rule of integration is a basic yet essential tool for finding antiderivatives of power functions. It applies seamlessly to functions where the integrand is a power of \( x \).
The power rule formula is as follows:

• For \( n eq -1 \), \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

Using this rule, you can integrate polynomials term by term. Each term of the polynomial can be treated as its own integral.
In your example, you applied the power rule to the term \( \int x \, dx \). Since \( \int x \equiv x^1 \), using \( n = 1 \) in the formula leads to \( \frac{x^2}{2} + C \). This straightforward application of the power rule significantly simplifies polynomial integration tasks.